94 SCHOOL SCIENCE AND MATHEMATICS

result, many nights my father would work late in the shop and myjob would be to hold the light. When the shoes were fitted and readyto nail he would say to me as he drew up the shoeing box, "Now,Bayard, you hold the light so that you can see, then I can see too."

I am sure that one of the primary objectives in your educationalwork is to have "good light" for yourselves and to hold it so thatthe boys and girls who are under your training and instruction"can see too."

WHAT IS MEANT BY THE AREA OF A CIRCLE?

BENJAMIN GREENBERGEastern District High School

Brooklyn) New York

H. P. Fawcett in developing the nature of proof came face to face with theproblem of definition. He felt, and most of us agree, that a good definition ofsome concept must essentially mean the same to the various people who studythe definition.A satisfactory definition of a circle should have a meaning, clear and precise

and should mean the same thing to all people looking at the definition. A surveyof numerous textbooks in plane geometry reveals two definitions. The firstdefinition considers a circle as the bounding curve and the area within thecurve. (We will refer to this as definition one.) The other definition considersthe circle as the curve alone. Nothing is said about the space inside the curve.(We will refer to this as definition two.)As a matter of interest many plane geometry textbooks were consulted in

order to determine whether definition one or definition two was employed. It isinteresting to note that prior to 1923 definition one was utilized with an occasionalreference to definition two. After 1923 definition two has been utilized almostwithout exception. This is not too strange when one considers that in 1923 thefamous report of the National Committee on Mathematical Requirements waspublished. They wanted a circle to be defined as the curve, not the curve andthe space enclosed by the curve. This report has influenced the writing of text-books since that time and so it isn’t surprising that plane geometry textbookssince that time have utilized the definition of a circle as recommended by theNational Committee on Mathematical Requirements.Another important influence on mathematics, namely the fifteenth yearbook

of the National Council of Teachers of Mathematics, also defines a circle as thecurve and not the space enclosed by the curve.

It should be mentioned that F. W. A. Young advocated the definition of thecircle as the curve in 1907.Thus far this bit of research has merely uncovered the fact that an important

committee often can unify thought,on a particular topic or definition. The mainreason this article has been written is to point out that most writers of planegeometry textbooks have defined a circle as a curve, but then have gone on tospeak of the area of a circle. Strictly speaking there is no such thing as the areaof a circle since a circle is defined as a curve and not as the figure enclosed bythe curve. When the authors of the textbooks who define the circle as the curvecome to the topic of "areas" of circles they should point out in their textbooksthat there is no such thing as the area of a circle, but what is wanted is the areaof the figure enclosed by the circle. After this has been specifically mentioned inthe textbook, then the author should point out that thereafter when he speaksof the "area" of a circle he actually means the area enclosed by the circle.